In the paper "Hasse principle" for $PSL_2 (\mathbb Z)$ and $PSL_2(\mathbb F)$ there's a definition of a Hasse principle for a group $G$, but I don't completely get it. Is there a more motivated reformulation of this definition?

Why I am interested: local-global principles are often very interesting in arithmetic geometry, so when I noticed a paper with this title I looked at it to see whether this proves something geometric.

As said below, one can formulate a problem of computing a group $Sha$ defined by a $g$-module $G$ and a family of subgroups $h_i\in G$, but the actual computation in the paper is for a specific choice of $h_i$, and I can't parse if there is an application of interest. Is it so?

(I suspect this problem arises when you try to prove Hasse principle for equations, like $x^n = a$ but with different Galois groups, see their first paper, though the results there have a mistake, corrected in the next one)